How does one prove existence decomposition of $R^2$ for countable many subsets $A(n)$ s .t.$\forall$ $x,y$ $\epsilon$ $A(n)$ $|x-y|$ is nonrational? I tried thinking of $R^2$ as …

Consider the following game, played by two players, called Q and A, in a time frame t = 1, 2, .... At every time point i, Q mentions some $Q_i \subset \mathbb{R}$, after which A m …

Is the following true? If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $( …

For $m,n,k < \omega$, consider the equation $X \to (\omega \times k)^{m}_{n}$ What is the smallest $X$ known to satisfy it? Baumgartner-Hajnal theorem gives a satisfactory an …

Defining a component as a maximally connected subgraph, is there a way to prove (or a counter-example) that every vertex belongs to some component, even in the case of infinite gra …

Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all suppo …